We plot the curves of the Chern numbers of bands 2 and 3 as a function of
the magnetic field in Fig. REF . The phonon Hall conductivity at
\(T=50 K\)  is also shown for comparison. To calculate the integer Chern numbers,
large number of \({\bf k}\) -sampling points \(N\)  is needed. However there is always a zero
eigenvalue at the \({\bf \Gamma }\)  point of the dispersion relation, which corresponds to
a singularity of the Berry curvature. Therefore, we cannot sum up
the Berry curvature very near this point to obtain Chern number of
this band, unless we add a negligible on-site potential
\(\frac{1}{2}u^TV_{\mathrm {onsite}}u\)  to the original Hamiltonian
[1]}, which will not change the topology of the space of
the eigenvectors. In Fig. REF , we set
\(V_{\mathrm {onsite}}=10^{-3} K_L\) . The Chern numbers of
bands 2 and 3 have three jumps with the increasing of the magnetic
field, although the phonon Hall conductivity is continuous.
For other bands, the Chern numbers keep constant: \(C^1=C^4=-1\) , \(C^5=0\) ,
and \(C^6=1\) . For the electronic Hall effect, we know it is quantized because
the Hall conductivity is directly proportional to the quantized Chern numbers.
Here we also find the quantized effect of the Chern numbers from
Fig. REF , while there is no quantized effect for the phonon
Hall conductivity. Such difference of the PHE from the electronic Hall
effect comes from the different nature of the phonons respective to the
electrons. In Eq. (REF ), in the summation, an extra term
\( (\omega _\sigma + \omega _{\sigma ^{\prime }})^2 \)  relating to the phonon
energy which is an analog of the electrical charge term \(e^2\)  in the
electron Hall effect, can not be moved out from the summation. Combining
the Bose distribution, the term \(f(\omega _\sigma ) (\omega _\sigma +\omega _{\sigma ^{\prime }})^2 \)  make the phonon Hall conductivity smooth,
no discontinuity comes out although the Chern numbers have some
sudden jumps. From the discussion in Ref. [1]},
the discontinuity of the Chern numbers corresponds to the phase transitions
and would relate to the divergency of derivative of the phonon Hall conductivity.
<FIGURE><FIGURE>